chol {base} | R Documentation

## The Choleski Decomposition

### Description

Compute the Choleski factorization of a real symmetric positive-definite square matrix.

### Usage

    chol(x, pivot = FALSE,  LINPACK = pivot)
    La.chol(x)

### Arguments

`x` | a real symmetric, positive-definite matrix
`pivot` | Should pivoting be used?
`LINPACK` | logical. Should LINPACK be used (for compatibility with R < 1.7.0)?

### Details

`chol(pivot = TRUE)` provides an interface to the LINPACK routine DCHDC. `La.chol` provides an interface to the LAPACK routine DPOTRF.

Note that only the upper triangular part of `x` is used, so that R'R = x when `x` is symmetric.

If `pivot = FALSE` and `x` is not non-negative definite an error occurs. If `x` is positive semi-definite (i.e., some zero eigenvalues) an error will also occur, as a numerical tolerance is used.

If `pivot = TRUE`, then the Choleski decomposition of a positive semi-definite `x` can be computed. The rank of `x` is returned as `attr(Q, "rank")`, subject to numerical errors. The pivot is returned as `attr(Q, "pivot")`. It is no longer the case that `t(Q) %*% Q` equals `x`. However, setting `pivot <- attr(Q, "pivot")` and `oo <- order(pivot)`, it is true that `t(Q[, oo]) %*% Q[, oo]` equals `x`, or, alternatively, `t(Q) %*% Q` equals `x[pivot, pivot]`. See the examples.

### Value

The upper triangular factor of the Choleski decomposition, i.e., the matrix R such that R'R = x (see example).
If pivoting is used, then two additional attributes `"pivot"` and `"rank"` are also returned.

### Warning

The code does not check for symmetry.

If `pivot = TRUE` and `x` is not non-negative definite then there will be no error message but a meaningless result will occur. So only use `pivot = TRUE` when `x` is non-negative definite by construction.

### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978) LINPACK Users Guide. Philadelphia: SIAM Publications.

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at http://www.netlib.org/lapack/lug/lapack_lug.html.

### See Also

`chol2inv` for its inverse (without pivoting), `backsolve` for solving linear systems with upper triangular left sides.

`qr`, `svd` for related matrix factorizations.

### Examples

    ( m <- matrix(c(5,1,1,3),2,2) )
    ( cm <- chol(m) )
    t(cm) %*% cm  #-- = 'm'
    crossprod(cm)  #-- = 'm'

    # now for something positive semi-definite
    x <- matrix(c(1:5, (1:5)^2), 5, 2)
    x <- cbind(x, x[, 1] + 3*x[, 2])
    m <- crossprod(x)
    qr(m)$rank # is 2, as it should be

    # chol() may fail, depending on numerical rounding:
    # chol() unlike qr() does not use a tolerance.
    try(chol(m))

    (Q <- chol(m, pivot = TRUE)) # NB wrong rank here ... see Warning section.
    ## we can use this by
    pivot <- attr(Q, "pivot")
    oo <- order(pivot)
    t(Q[, oo]) %*% Q[, oo] # recover m

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